First Order Analysis of Uncertainty in Numerical Modelsof Groundwater Flow Part 1. Mathematical Development

In part 1 of this paper, first and second order analysis of uncertainty is applied to numerical models of groundwater flow. The models are cast in state-space form, with boundary conditions and inputs that are functionally dependent, but statistically independent, of time. Using a compact matrix calculus notation, first and second order Taylor series expansions of the model equations are derived and used to estimate the mean and variance-covariance properties of piezometric head predictions, given corresponding statistics for aquifer parameters: material properties, initial conditions, boundary conditions, and inputs. The mathematical results demonstrate that the prediction uncertainty is a function of the magnitude of the parameter uncertainty, and sensitivity of the predictions to the parameters. Furthermore, the first order estimate of the piezometric head is identical to the dete/'ministic result. Part 2 of this paper, to be presented later, will illustrate these and other results through numerous applications of the methodology.